Triangles, Numbers, and
Snowflakes!
-Mathematical
Reasoning
-Geometry
-Number
Sense
-Representations
-Patterns,
Relationships, and Functions
Purpose: The purpose of this lesson
is to explore patterns in geometry and to discover how those patterns are all
related to each other.
Materials:
-Movie: Fractals and the
Mandelbrot Set
-Activity
Sheets
-Pencil
-Computer
Lab
Time
Required:
1-2 days or more if students are really getting into the applets!
Lesson
Procedure:
I began my lesson by introducing fractals and patterns to my students. We
discussed all of the patterns in nature that we could think of and how some of
the patterns could go on forever if they wanted to. We also talked about how
some patterns are perfect and how they could possibly get that way.
Next
I showed students the beginning of a movie that I thought would stir up some
imagination for them. The movie introduces Mandelbrot and his theories. The
movie also shows the Mandelbrot set in Technicolor with great music! We only
watch about 20 minutes of it because it becomes too heavy for 6th
graders to digest. It is a great introduction for them to see deeper and deeper
into the set.
When
the movie is over I introduce the “Sierpinski Triangle’ and “Koch Snowflake”
worksheets to students and tell them that they are going to create their own
fractals by following the instructions in the page. I provide the students with
plenty of extra erasers and triangular grid sheets for their mistakes. I also
ask the students to complete the questions on their worksheet on a separate
sheet of paper to be turned in to me with their completed triangles and
snowflakes. This activity usually consumes 2 days of instructional time but is
well worth it.
On
day 2 I introduce “Pascal’s Triangle” and throw another wrench into the fun.
Now the students will need to follow the directions on the triangle page to
complete. On the back of that page there is an accompanying math question
worksheet that students will need to fill out.
Day
3: Students will share their triangles and snowflakes with the class and what
they learned while constructing each. The class will also share their answers
to the Sierpinski triangle questions and reveal the formulas and patterns they
have come up with.
Explorations
and Extensions:
http://mathforum.org/student/showcase/jeff.jrhigh//
: Fractals and Tessellations student projects on the web. COOL!
http://math.rice.edu/~lanius/frac/
: Fractals and Sierpinski Triangle links!
Assesment
and Evaluation tools:
Look
at student created triangles and snowflakes.
Give
a warm-up/pop quiz question asking students to describe the patterns in the
triangles/snowflakes they have created and how they relate to each other.
Content
Standards:
Standard
3: Geometry
Standard
6: Problem Solving
Standard
7: Mathematical Reasoning and Proof
Standard
8: Mathematical Communication
Standard
10: Mathematical Representations
Personal
Comments:
I
had my students do this lesson last week. They thoroughly enjoyed drawing the
geometric patterns and finding the next values in Pascal’s Triangle. They were
able to get a basic understanding of the relationship between the two. They had
a very difficult time answering some of the questions in the packet, especially
trying to find a formula for the “next”….
Peer
feedback: